A294539 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + n - 1, where a(0) = 1, a(1) = 2, b(0) = 3.

1, 2, 7, 15, 30, 55, 98, 168, 283, 470, 774, 1267, 2066, 3361, 5457, 8850, 14341, 23227, 37606, 60873, 98521, 159438, 258005, 417491, 675546, 1093089, 1768689, 2861835, 4630583, 7492479, 12123125, 19615669, 31738861, 51354599, 83093531, 134448203

Offset: 0

Author

Clark Kimberling, Nov 03 2017

Keywords

• nonn
• easy

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294532 for a guide to related sequences. Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622)..

Examples

			a(0) = 1, a(1) = 2, b(0) = 3, so that
b(1) = 4 (least "new number")
a(2)  = a(1) + a(0) + b(0) + 1 = 7
Complement: (b(n)) = (3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 16, ...)
		

Links

• Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Crossrefs

Cf. A001622, A294532.

Programs

Mathematica

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 2] + n - 1;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A294539 *)
Table[b[n], {n, 0, 10}]